Should I replace recirc pump for my tankless heater?

Hi-
I recently bought a house which has a tankless heater, and is coupled with a recirc pump which isn't working. The system does have a dedicated water line for the recirc pump. Unfortunately the designer of the house put the water heater at absolutely the opposite end of the house from where all the water is used, and consequently it takes 1-2 minutes of running the water before hot water shows up when the pump isn't working. I have a few questions about it:
1) If I replace the pump, how fast will hot water get to the outlets? It seems like theoretically it could be almost instant, is that right?
2) With a pump in operation, is there a risk of having the water go warm/cold/hot, or are temperature fluctuations few/gradual?
3) Isn't the whole purpose of an "on demand" water heater nullified by using a recirc pump, or is it actually still more efficient than a tank heater?
4) How hard is it to swap out a broken pump... I've done a little bit of soldering etc. before, but am no plumber, that's for sure.

In case you're wondering, my tankless is a Noritz N-084M-DV and my recirc pump is a Laing Ultracirc UCT-909.

It somewhat depends on how many return lines you have and how well they are balanced as to how fast you get hot water, but, yes, done right, it is almost instantaneous. Some tankless systems should not be run with a recirculation system, but if you are allowed, then there are specific rules on how it is to be done. You may need to add a small buffer tank to do it right and not have the heater running all the time there's water circulating.

Replacing pumps may be easy and not require any soldering at all, depends on how it was installed. On some, there are gasketed connections, and you just unbolt the thing. Assuming you get the same pump, a new one should bolt in (maybe with new gaskets). The electrical connection normally requires a screwdriver and maybe some wire nuts.

I have been looking at the recirc system with an eye toward just swapping out the pump (it does look like it is just screwed in, not soldered), and I was also looking at the install manual for Laing pumps, and noticed a note "the pump must always be installed below the water heater...". Well, in the installation in my house the pump is above the tankless heater. Does this matter? I can also see that interestingly enough, the section of return pipe that the pump is measuring temperature off of is warm to the touch, as well as the pump body itself, so it does seem like warm water is finding its way into the pump, unless it could be overheating itself. Additionally, the connections to the pump are with 1/2" pipes, even though the manual specifically says to use a minimum of 3/4" pipe to get optimal flow through the tankless heater. All of this is making me wonder if the pump was just not installed properly to begin with and may never have worked. Is there an easy way to determine whether or not the pump is actually pumping water?

I never did get this figured out, and now that winter is coming I'm thinking about it again Any suggestions on what I should try? Or failing that, a recommendation on a plumber with experience troubleshooting tankless recirc issues in the Seattle, WA area?

The half inch pipe isn't likely the problem- the pump should be able to drive well over the minimum flow for lighting off the tankless even through 100' of half inch pipe, but the location of the pump above the HW heater and adjacent to the exhaust might be responsible for keeping it higher than anticipated sensed water temp that could inhibit it from running sometimes. If you can cool the pump and adjacent with ice or a fan to where it's cool to the touch and the pump doesn't turn on either the pump or the controls are toast.